Pendulums and predictability
Before you start
A pendulum clock counts repeated motion: the bob swings back and forth, letting the mechanism keep going. For the reading to be reliable, each return must take a sufficiently steady time. Like the object on a spring, the bob passes through the position it is drawn back towards and returns again. Here it hangs below a pivot and moves along an arc. The pendulum length is the distance from that pivot to the bob’s centre. We can change this distance ourselves and see how the clock’s rhythm changes with it.
Watch how the clock advances when the bob passes its lowest point, then drag the bob aside and release it. When comparing lengths, use the same small starting angle so that the change in rhythm is easier to connect to length. Let several swings pass and watch a small timing difference accumulate on the clock. The double pendulum later asks a different question: if two starting angles are almost identical, must the later motions stay close too? We will keep the motions together so that the developing difference can be seen.
The setupA regulator clock hangs on the wall with its brass pendulum bob. The hands start at your device’s local time, and each time the bob passes its lowest point, the second hand advances by one second. You can drag the bob with a mouse or finger and release it, and change the pendulum length; the default, about 0.994 m, gives one beat per second. The double-pendulum view shows two separate pairs of silver bobs joined by light rods, started at almost the same angle.
What to do and noticeWatch the bob and the second hand together until you can predict the next tick. Then drag the bob to one side, release it, and count the beats against a real clock. Next, lengthen the pendulum and count again: every beat still adds one second to the clock face, but each beat now takes longer in real time, so the clock falls behind. In the double-pendulum view, the two pairs start only 0.1° apart. Watch whether that small difference stays small.
Predict firstWhere is the bob when the second hand advances?
How can a clock count time using a swing? Start a regulator from your local time, change its pendulum length, then compare two almost identical double pendulums.
Prerequisites: Length is the distance from the pivot to the bob’s centre. One beat is one pass through the lowest point; a full return is two beats. Angles are measured from the downward vertical. Drag the brass pendulum bob with a mouse or finger, then release it. The case and room are fixed scenery. Refreshing starts the hands at your device’s local time; each passage through the lowest point advances the second hand by one second. The default length is about 0.994 m. The double-pendulum view compares two independent pairs of silver bobs, joined by light rods.How a pendulum counts seconds
Predict firstWhere is the bob when the second hand advances?
First look at the dial. On refresh, the three hands start from your device’s local time. After that, the clock counts its own beats instead of continually synchronising. Every crossing of the lowest point moves the second hand one division. At the default length of about 0.994 m, a swing from one side to the other takes about 1 s, and the return takes another 1 s. A full return is the period , about 2 s. One beat is one pass through the lowest point, so a beat is T/2, about 1 s.
Drag the brass bob aside and release it. The tangential component of gravity pulls it towards the bottom. It arrives with speed, so it continues up the other side; the rod constrains its circular path. While you hold it, no new crossing occurs and the clock stops counting. Release it to resume. We measure angle from the downward vertical.
How to observe
- Where is the bob when the second hand advances?
- Why does the clock slow if every beat still advances one second?
- Will a 0.1° difference remain tiny after 20 seconds?
What this model simplifies
The bobs are ideal point masses on light rods, with gravitational acceleration 9.81 metres per second squared; resistance and energy lost to clockwork are neglected. A real clock needs an escapement to replace these losses. Clock dragging is limited to 20° either side of vertical. Motion keeps the full sine of the angle, so larger swings take slightly longer than the small-angle formula predicts. Not all double-pendulum motions are strongly chaotic.
